Vol. 14, No. 4, 2021

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Multilayer potentials for higher-order systems in rough domains

Gustavo Hoepfner, Paulo Liboni, Dorina Mitrea, Irina Mitrea and Marius Mitrea

Vol. 14 (2021), No. 4, 1233–1308
Abstract

We initiate the study of multilayer potential operators associated with any given homogeneous constant-coefficient higher-order elliptic system L in an open set Ω n satisfying additional assumptions of a geometric measure theoretic nature. We develop a Calderón–Zygmund-type theory for this brand of singular integral operators acting on Whitney arrays, starting with the case when Ω is merely of locally finite perimeter and then progressively strengthening the hypotheses by ultimately assuming that Ω is a uniformly rectifiable domain (which is the optimal setting where singular integral operators of principal value type are known to be bounded on Lebesgue spaces), and conclude by indicating how this body of results is significant in the context of boundary value problems for the higher-order system L in such a domain Ω.

Keywords
higher-order system, multilayer operator, boundary layer potential, Calderón–Zygmund operator, principal value singular integral operator, set of locally finite perimeter, Ahlfors regular set, uniformly rectifiable set, divergence theorem, nontangential maximal operators, nontangential boundary trace, Whitney arrays, boundary Sobolev space, Carleson measure, Dirichlet boundary problem, regularity boundary problem
Mathematical Subject Classification 2010
Primary: 31B10, 31B25, 35C15, 42B20, 42B37, 49Q15
Secondary: 35A01, 35A02, 35J58, 42B25, 42B35, 42B35, 45E05, 45P05, 47B38, 47G10
Milestones
Received: 7 May 2019
Revised: 10 November 2019
Accepted: 20 December 2019
Published: 6 July 2021
Authors
Gustavo Hoepfner
Departamento de Matemática
Universidade Federal de São Carlos
São Carlos
Brazil
Paulo Liboni
Departamento de Matemática
Universidade Estadual de Londrina
Londrina
Brazil
Dorina Mitrea
Department of Mathematics
Baylor University
Waco, TX
United States
Irina Mitrea
Department of Mathematics
Temple University
Philadelphia, PA
United States
Marius Mitrea
Department of Mathematics
Baylor University
Waco, TX
United States